Annals of Functional Analysis

Kaplansky's and Michael's problems‎: ‎a survey

Jean Esterle

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‎I‎. ‎Kaplansky showed in 1947 that every submultiplicative norm $\Vert‎ . ‎\Vert$ on the algebra ${\mathcal C}(K)$ of complex--valued‎ ‎functions on an infinite compact space $K$ satisfies $\Vert f \Vert‎ ‎\ge \Vert f \Vert _K$ for every $f \in {\mathcal C}(K),$ where‎ ‎$\Vert f \Vert _K=max_{t \in K} \vert f(t)\vert $ denotes the‎ ‎standard norm on ${\mathcal C}(K).$ He asked whether all‎ ‎submultiplicative norms $\Vert‎ . ‎\Vert$ were in fact equivalent to‎ ‎the standard norm (which is obviously true for finite compact‎ ‎spaces)‎, ‎or equivalently‎, ‎whether all homomorphisms from ${\mathcal‎ ‎C}(K)$ into a Banach algebra were continuous‎. ‎This problem turned‎ ‎out to be undecidable in ZFC‎, ‎and we will discuss here some recent‎ ‎progress due to Pham and open questions concerning the structure of‎ ‎the set of nonmaximal prime ideals of ${\mathcal C}(K)$ which are‎ ‎closed with respect to a discontinuous submultiplicative norm on‎ ‎${\mathcal C}(K)$ when the continuum hypothesis is assumed‎. ‎We will‎ ‎also discuss the existence of discontinuous characters on Fr\'echet‎ ‎algebras (Michael's problem)‎, ‎a long standing problem which remains‎ ‎unsolved‎. ‎The Mittag--Leffler theorem on inverse limits of complete‎ ‎metric spaces plays an essential role in the literature concerning‎ ‎both problems‎.

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Ann. Funct. Anal., Volume 3, Number 2 (2012), 66-88.

First available in Project Euclid: 12 May 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46H40: Automatic continuity
Secondary: 32A10‎ ‎46J10‎ ‎46J45

Mittag-Leffler theorem ‎continuity of characters ‎Fr\'echet algebras ‎Banach algebras ‎continuum hypothesis


Esterle, Jean. Kaplansky's and Michael's problems‎: ‎a survey. Ann. Funct. Anal. 3 (2012), no. 2, 66--88. doi:10.15352/afa/1399899933.

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